[Paper Review] A New Lower Bound for the Random Offerer Mechanism in Bilateral Trade using AI-Guided Evolutionary Search

Yang Cai, Vineet Gupta+2 more|arXiv (Cornell University)|Mar 9, 2026

Auction Theory and Applications0 citations

TL;DR

The paper uses AI-guided evolutionary search (AlphaEvolve) to discover a new worst-case distribution pair that establishes a tighter lower bound on the Random Offerer (RO) mechanism’s worst-case approximation ratio, achieving 2.0749 relative to the first-best gains from trade.

ABSTRACT

The celebrated Myerson--Satterthwaite theorem shows that in bilateral trade, no mechanism can be simultaneously fully efficient, Bayesian incentive compatible (BIC), and budget balanced (BB). This naturally raises the question of how closely the gains from trade (GFT) achievable by a BIC and BB mechanism can approximate the first-best (fully efficient) benchmark. The optimal BIC and BB mechanism is typically complex and highly distribution-dependent, making it difficult to characterize directly. Consequently, much of the literature analyzes simpler mechanisms such as the Random-Offerer (RO) mechanism and establishes constant-factor guarantees relative to the first-best GFT. An important open question concerns the worst-case performance of the RO mechanism relative to first-best (FB) efficiency. While it was originally hypothesized that the approximation ratio $\frac{ ext{GFT}_{ ext{FB}}}{ ext{GFT}_{ ext{RO}}}$ is bounded by $2$, recent work provided counterexamples to this conjecture: Cai et al. proved that the ratio can be strictly larger than $2$, and Babaioff et al. exhibited an explicit example with ratio approximately $2.02$. In this work, we employ AlphaEvolve, an AI-guided evolutionary search framework, to explore the space of value distributions. We identify a new worst-case instance that yields an improved lower bound of $\frac{ ext{GFT}_{ ext{FB}}}{ ext{GFT}_{ ext{RO}}} \ge extbf{2.0749}$. This establishes a new lower bound on the worst-case performance of the Random-Offerer mechanism, demonstrating a wider efficiency gap than previously known.

Motivation & Objective

• Motivate the study of how closely RO and BB/BIC mechanisms can approximate the first-best gains from trade in bilateral trade.
• Identify worst-case buyer-seller value distributions that maximize the efficiency gap for RO.
• Leverage AI-guided evolution to discover novel distribution structures that maximize GFT gaps.
• Provide a rigorous, exact computation of GFT under discretized distributions to certify the lower bound.
• Demonstrate the potential of AI-driven program synthesis for revealing non-intuitive mechanism-design results.

Proposed method

• Formulate the problem as maximizing the ratio of First-Best GFT to RO GFT over distributions (F_s, F_b).
• Fix the buyer distribution to Discrete Equal Revenue and evolve the seller distribution F_s using AlphaEvolve."
• Represent the seller’s CDF as a mixture of modulated power laws with sinusoidal exponent modulation.
• Discretize the domain to H = 20,000 and compute GFT exactly using integer arithmetic with high precision.
• Round PMFs to multiples of ε = 1e-15 to avoid floating-point errors and ensure exact GFT calculations.
• Report the resulting worst-case ratio and the corresponding GFT components.

Figure 1: Seller’s distribution as found by AlphaEvolve

Experimental results

Research questions

• RQ1 What is the worst-case approximation ratio ρ = GFT_FB / GFT_RO over distributions of seller cost and buyer value?
• RQ2 Can AI-guided evolutionary search discover seller distributions that yield a larger GFT gap than previously known?
• RQ3 What is the structure of the seller distribution that maximizes the RO inefficiency under a fixed buyer distribution?
• RQ4 How does the discovered distribution affect the relative contributions of Seller-Offering and Buyer-Offering GFT components?
• RQ5 Is the new lower bound of 2.0749 robust under discretization and exact computation?

Key findings

• A new worst-case approximation ratio of 2.0749 is achieved for the RO mechanism.
• The configuration uses a fixed buyer distribution (Discrete Equal Revenue) paired with a novel evolved seller distribution.
• The seller distribution is a mixture of modulated power laws with a sinusoidally modulated exponent.
• GFT_FB ≈ 1.2322, GFT_SO ≈ 0.3312, GFT_BO ≈ 0.8565, GFT_RO ≈ 0.5939.
• This yields ρ = GFT_FB / GFT_RO ≈ 2.0749, beating the previous lower bound near 2.02.
• The evolved seller distribution employs a sinusoidal modulation parameter a1_amp = 0.05 and a1_freq = 2.0.